You have been asked to design a "ballistic spring system" to INT measure the speed of bullets. A bullet of mass \(m\) is fired into a block of mass \(M\). The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is \(k\). The opposite end of the spring is anchored to a wall. The spring's maximum compression \(d\) is measured.

a. Find an expression for the bullet's initial speed \(v_{\mathrm{B}}\) in terms of \(m, M, k\), and \(d\).

b. What was the speed of a \(5.0 \mathrm{~g}\) bullet if the block's mass is \(2.0 \mathrm{~kg}\) and if the spring, with \(k=50 \mathrm{~N} / \mathrm{m}\), was compressed by \(10 \mathrm{~cm}\) ?

c. What fraction of the bullet's initial kinetic energy is "lost"? Where did it go?